How to estimate total differential attenuation due to hydrometeors with ground-based multi-frequency radars?

At millimeter wavelengths, attenuation by hydrometeors, such as liquid droplets or large snowflakes, is generally not negligible. When using multi-frequency ground-based radar measurements, it is common practice to use the Rayleigh targets at cloud top as a reference in order to derive attenuation-corrected reflectivities and meaningful dual-frequency ratios (DFR). By capitalizing on this idea, this study describes a new quality-controlled approach aiming at identifying regions of the cloud where particle growth is negligible. The core of the method is the identification of a “Rayleigh plateau”, i.e. a large enough 5 region near cloud top where the vertical gradient of DFR remains small. By analyzing collocated Ka-W band radar and microwave radiometer (MWR) observations taken at two European sites under various meteorological conditions, it is shown how the resulting estimates of differential path-integrated attenuation (∆PIA) can be used to characterize hydrometeor properties. When the ∆PIA is predominantly produced by cloud liquid droplets, this technique alone can provide accurate estimates of the liquid water path. When combined with MWR observations, this 10 methodology paves the way towards profiling the cloud liquid water and/or quality flagging the MWR retrieval for rain/drizzle contamination and/or estimating the snow differential attenuation.

tain clutter (Delrieu et al., 1997;Serrar et al., 2000); 3. the Doppler spectral ratio techniques which require radar observations at multiple frequencies and are based on recovering differential attenuation profiles from the spectral power ratios of the Doppler velocities corresponding to the Rayleigh slow-falling particles (Tridon and Battaglia, 2015;Tridon et al., 2017a;Li and Moisseev, 2019).
This study focuses on the third method, i.e. the exploitation of the small targets that backscatter according to Rayleigh law 70 (Bohren and Huffman, 1983) at all radar observing frequencies as tracers of the differential path integrated attenuation ∆PIA.
However, this method can be applied with standard radar moments and does not require to record high-quality Doppler spectra.
Previously, the cloud region with potential Rayleigh particles has been identified using thresholds of reflectivity (Hogan et al., 2000;Kneifel et al., 2015;Dias Neto et al., 2019). Although there is undoubtedly a general correlation between the strength of differential scattering and radar reflectivity (Matrosov et al., 2019), this threshold method also has a number of disadvantages. 75 First, the threshold depends on frequency (lower frequency radars can accept a larger reflectivity threshold) and if a too strict threshold is chosen, the region with potential Rayleigh targets might become very small. In other situations, the concentration of larger particles might be small enough to cause a reflectivity smaller than the threshold, but their differential scattering signal might be non-negligible. The threshold method also does not apply any quality control on the differential reflectivities themselves, which are often found to be rather noisy (Battaglia et al., 2020b) due to non perfect backscattering volume matching 80 and possible mispointing of the antenna beams.
In this paper, a rigorous new procedure for deriving ∆PIA from ground-based multi-frequency radars is presented (description in Sect. 3) and exemplified in the case of the K a -W band pair of radar frequencies. It is then applied to a multi-layered cloud with an ice cloud on top of a low level liquid cloud (Sect. 4) and a mixed-phase cloud with supercooled liquid layers embedded in an ice cloud (Sect. 5). The impact and the potential benefits/applications of this technique are discussed in Sect. 6.
During the past decade, millimeter wavelength (cloud) radars have become essential tools for the observations of clouds and precipitation. Cloud radars provide particular advantages for cloud and precipitation studies due to their narrow beam width, inherent high sensitivity, portability, reduced susceptibility to Bragg scattering and ground clutter (Kollias et al., 2007). These advantages, however, come with the cost of larger signal attenuation caused by atmospheric gases and hydrometeors, which 90 in general increases with frequency. While attenuation mainly limits the maximum range of possible radar observations, the frequency dependent attenuation signal can also be used as source of information. For example, Hogan et al. (2005);Huang et al. (2009);Zhu et al. (2019) used the differential attenuation between K a and W band to infer cloud liquid water in pure liquid clouds. Similarly, the attenuation signal at K a band was used by Matrosov (2005); Matrosov et al. (2006) to derive rainfall rate. 95 For droplets and ice crystals whose sizes remain much smaller than wavelength of millimeter microwave radiation, the Rayleigh approximation (Bohren and Huffman, 1983) is applicable for computing scattering and absorption properties. In this regime, absorption and scattering efficiencies are both related to the size parameter, x ≡ πD λ (D is a characteristic size of the target, λ the wavelength of the radiation transmitted by the radar), i.e. proportional to x or x 4 , respectively. But, while absorption is proportional to the imaginary part of the Clausius-Mossotti factor K = n 2 −1 n 2 +2 (where n is the refractive index of 100 the scatterer), scattering is proportional to the absolute value of the square of K. The much larger imaginary part of the liquid water versus the ice refractive index explains the generally larger absorption of microwave radiation by liquid hydrometeors as compared to ice particles. Attenuation coefficients, defined as the integral of the absorption cross sections (efficiencies times π 4 D 2 ) over the PSD, are then proportional to the equivalent water mass per unit volume. On the other hand, large raindrops/ice crystals, graupel, and hailstones must be generally considered as non-Rayleigh targets 105 at millimeter wavelengths. In first approximation, by treating particles as spheres, electromagnetic scattering computations based on Mie theory can be used (Lhermitte, 1990). More complex computations are generally needed for accurately describing the scattering properties of large raindrops and snowflakes that exhibit non-spherical shapes (an exhaustive review is provided in Kneifel et al. (2020)).

Liquid hydrometeors 110
The attenuation coefficients per unit mass (hereafter indicated with k em and expressed in dB m 2 kg −1 ) for raindrops is shown in Fig. 2 of Battaglia et al. (2014). The starting value at small sizes corresponds to the Rayleigh absorption value for cloud water: The resulting one-way attenuation produced by a 1 km thick liquid cloud with a liquid water content of 1 g m −3 (corresponding 115 to a liquid water path of 1 kg m −2 ) is negligible at K u -band and becomes increasingly significant for larger frequencies, up to ≈ 4 dB km −1 at W-band (Fig. 1). The dielectric properties (i.e., refractive indices) of liquid water also depend on temperature. The different line types show the discrepancies between three recent models for computing the liquid water refractive index (Turner et al., 2016;Rosenkranz, 2015;Ellison, 2007).
Because laboratory measurements of the refractive index of supercooled liquid water are challenging, refractive index models differ more at negative temperatures. This is illustrated for three recent models (Ellison, 2007;Rosenkranz, 2015;Turner et al., 2016) shown in Fig. 1.

120
For rain (larger drop sizes), the attenuation coefficient steadily increases till a maximum close to D/λ ≈ 3, and then monotonically decreases to a frequency-independent value corresponding to extinction efficiencies of 2 as expected in the geometrical optics limit. Note that the contribution of scattering to attenuation increases with droplet size.

Solid hydrometeors
For ice particles, attenuation is dominated by scattering and is a steadily increasing function of size and density (see Fig. A6 125 in Battaglia et al. (2020a)). While attenuation by ice crystals is negligible at all frequencies below 200 GHz, snowflakes tend to produce non negligible attenuation at and above W-band. Recent findings by Protat et al. (2019) show that W-band one-way attenuation of the order of 0.5-0.8 dB km −1 for reflectivities between 13 and 18 dBZ and up to 2 dB km −1 for reflectivities exceeding 20 dBZ can be expected in the ice anvils of tropical convective clouds. Snow attenuation considerably increases when moving from the W to the G-band (Battaglia et al., 2014;Wallace, 1988), reaching one-way values of 0.9, 2.5 and 130 8.7 dB m 2 kg −1 at 96, 140, 225 GHz (Nemarich et al., 1988). Graupel (hail) particles found in deep convection produce tangible attenuation already at K a -band (K u -band) as demonstrated in Battaglia et al. (2016).

Melting hydrometeors
Melting particles are generally very efficient in attenuating microwave radiation because they tend to appear as large water particles so that the melting layer not only corresponds to a region of enhanced backscattering (bright band) but also of enhanced 135 extinction , which can account for several dBs of attenuation. Recent observational findings by Li and Moisseev (2019) refined parametrisations of melting layer attenuation at K a and W-band based on theoretical computations (Matrosov, 2008). As a reference, an equivalent 1 mm h −1 (3 mm h −1 ) rainfall is expected to produce an average one-way specific attenuation of 1.2 (1.9) at K a -band and 3.4 (4.7) dB km −1 at W-band during melting. Once melted completely into rain, the attenuation reduces to 0.2 (0.68) at K a -band and 1.43 (3.0) dB km −1 .

The DFR Rayleigh plateau method
By definition, the PIA monotonically increases with range, or remains constant if the hydrometeor attenuation is negligible.
Attenuation leads to a reduction of the measured radar reflectivity and cannot be easily estimated when using a single frequency radar. Since the attenuation coefficient generally increases with frequency, coincident measurements with an additional radar at a non-attenuating (or less attenuating) frequency provides a reference for determining the ∆PIA between the two frequencies. 145 When comparing the reflectivities measured (indicated with Z m ) by two radars operating at different frequencies f 1 and f 2 (f 1 < f 2 ), their difference in logarithmic units (expressed in dB and called the dual frequency ratios, DFR(f 1 , f 2 , r)) receives contribution from differential (non-Rayleigh) scattering and differential attenuation (Tridon et al., 2013;Battaglia et al., 2020a): = Z e,f1 (r) − Z e,f2 (r) non-Rayleigh where Z e is the effective reflectivity and EWC is the equivalent water content. In this work, the contribution of interest is the differential attenuation (second term of Equation

155
A simple traditional approach to ensure the presence of only small hydrometeors is to restrict the data to regions where the reflectivity is lower than a certain threshold (for example Z e < −10 dBZ in Dias Neto et al. (2019)). This is based on the assumption that increasing reflectivity is connected to growth processes and hence the presence of increasingly large particles.
While this is generally true, it is quite obvious that also a high concentration of small and hence perfect Rayleigh scatterers could produce a Z e which is larger than this threshold. An opposite scenario would be a very low number of snow aggregates 160 whose Z e would be below the threshold but the snowflakes would be far from being Rayleigh targets. From those examples, 6 https://doi.org/10.5194/amt-2020-159 Preprint. Discussion started: 5 May 2020 c Author(s) 2020. CC BY 4.0 License.
the problem with a fixed Z e threshold becomes obvious. The threshold does not only depend on the radar frequency, but also on the details of the particle size distribution, and hence, in principle, has to be adjusted on a case by case basis.
In this work, a more general approach is proposed for estimating ∆PIA from zenith-pointing multi-frequency radar measurements. The DFR is closely related to the characteristic size of the PSD, which is generally expected to increase due to 165 particle growth processes. A DFR threshold would therefore seem to be a more reliable measure for the presence of large non-Rayleigh targets. However, larger DFRs could be also caused by attenuation due to gases and hydrometeors from layers below. As discussed before, Doppler spectra principally allow to disentangle attenuation and scattering effects. However, the spectra at cloud top are in general very narrow, which makes the separation more challenging. Also antenna mispointing effects (different shift of spectra) can be expected to be maximum at high altitudes due to the general stronger horizontal winds.

170
In the new approach, ice particles close to cloud top are assumed to be small enough to produce negligible differential scattering and attenuation. As a result, any measured DFR should be a result of path integrated attenuation from the cloud below. When moving downward from cloud top, the DFR remains constant down to the altitude where some ice particles reach sizes which cause non-Rayleigh scattering at the highest frequency used. The goal of this method is therefore to find a plateau of DFR close to cloud top and will be denominated the "DFR Rayleigh plateau method". Its advantage is that the potential 175 presence of few large aggregates, which can deteriorate the PIA estimate, will be detected by the DFR plateau approach even if all reflectivities in the layer were below the Z e threshold.
A gradient in DFR can also be caused by, for example, attenuation due to a layer of liquid water. However, in this case, the DFR increases with height, which is in general opposite to the growth of ice particles towards the ground. Hogan et al. (2005) used the DFR gradient in liquid stratocumulus clouds to derive liquid water content profiles. They mention that the DFR profiles 180 must be substantially averaged (they use 1 min and 150 m resolution) before one can exploit the few dBs variation produced by liquid attenuation. Indeed, the DFR profiles can become very noisy due to the random error of reflectivity measurements, especially in case of low signal-to-noise-ratios which can be encountered at cloud tops. Additional noisiness of the DFR signal can be caused by the potential mismatch between multi-frequency antenna beams in regions of strong spatial inhomogeneities.
Specifically, another advantage of this technique is that the presence of a DFR Rayleigh plateau is an indication of the good 185 quality of the radar beam alignment. For example, even in a horizontally homogeneous cloud, mispointing could lead to a perceptible DFR gradient. Therefore a rigorous procedure, hinged upon the identification of the DFR plateau at cloud top, is required to derive ∆PIA and assess its quality.

Description of the ∆PIA derivation
The major steps of the DFR Rayleigh plateau method are synthesized in the flow chart of Fig. 2. The approach can be applied 190 to any hydrometeor type, as long as it produces enough differential attenuation for the considered radar frequency pair. In the following, it is exemplified for the K a -W band pair.
The procedure for DFR profiles processing can be divided in two streams. The main stream (left part of  Parallelogram, diamond and rectangle represent input/output, task and decision, respectively.
determines the Rayleigh plateau region within the profile. The final ∆PIA is then derived from the filtered DFR which falls 195 into the plateau region.
In the main stream, ∆PIA is derived by taking the median of the DFR profile after it has been successively filtered using the following criteria: similarly to Dias Neto et al. (2019), the DFR variance within 20 s by 150 m moving windows must be lower than 4 dB 2 in order to remove the cloud regions potentially affected by a mismatch of the two radar beams;

200
-Z m,f1 and its variance (within the same 20 s by 150 m moving windows) must be lower than Z threshold1 (5 dBZ at K a -band) and 2.5 dB 2 , respectively, in order to exclude the regions where non-Rayleigh scattering is very likely, and where the cloud is highly turbulent; only the part of the profile, which has been identified as a Rayleigh plateau, is retained.  except that a finer time resolution is achieved. In order to limit spurious local variations, a fit of the DFR profile is obtained via a Savitzky-Golay filter (Orfanidis, 1996); DFR plateau regions are then defined as portions of the profile for which the absolute value of the DFR gradient is lower than 1 dB km −1 . Finally, a plateau is confirmed as a Rayleigh plateau only if it has a minimum thickness of 200 m and if it is located less than 500 m from cloud top.

Application to example profiles
The methodology is illustrated in Figs. 3 with two profiles taken from the case study presented in Section 5 for which subfreezing temperatures were recorded at the ground.
The profile in panels (a,b) of Fig. 3 depicts two separated cloud layers. The DFR continuously increases in the lower layer (0 to 3 km), while being nearly constant at around 6 dB in the upper cirrus cloud (6 to 8 km). The very low reflectivities (lower 215 than -10 dBZ) (see Fig. 9a) indicate that the cirrus is most likely composed of small ice crystals which do not produce any differential scattering signal. This is confirmed by the clear Rayleigh plateau (green shading) in this layer and therefore the DFR in this region can be used for estimating ∆PIA. The fact that the DFR is constant through the whole cirrus cloud indicates that the attenuation must be produced by the lower cloud layer. As will be shown in section 5, the attenuation is caused by a mixed-phase cloud with a total of 6 dB ∆PIA produced by a considerable amount of supercooled liquid water. Panels (c,d) of Fig. 3 shows a profile measured a couple of hours later with the DFR reaching values up to 9 dB before decreasing to about 5 dB. Since the differential attenuation must increase with range, such a decrease can only be explained by a reduction of the differential scattering, indicating that the cloud layer between 2 and 4 km is composed of large snowflakes.
Nevertheless, the upper cirrus cloud appears to be still composed of small ice crystals and the Rayleigh plateau identified between 6 and 8 km suggests that a total attenuation of 5 dB is produced by the lower cloud.

225
In the following case studies, this procedure is applied to consecutive DFR profiles, thus providing the time evolution of ∆PIA. . However, in this latter dataset, the DFR was calibrated using the traditional Ze-threshold approach i.e., by determining a relative offset between K a and W-band reflectivity within 15 min 235 time windows for Z Ka < −10 dBZ. Since the current study aims at refining such a procedure, the uncalibrated DFR is first recovered by subtracting this offset from the W-band reflectivity.
A thick ice cloud connected to a cold front was slowly moving over JOYCE on this day with only very weak precipitation (less than 1 mm total accumulation) starting around 11 UTC (Fig. 4a). The DFR reveals growth of larger snow particles starting at temperatures warmer than -15°C (Fig. 4b). In addition to the spatially inhomogeneous DFR structures related to differential 240 scattering, vertical lines of enhanced DFR up to cloud top indicating significant differential attenuation during the precipitating period (e.g. around noon) are visible as well.

Attenuation due to pure liquid cloud
In order to test the DFR Rayleigh plateau method for deriving ∆PIA, it is desirable to find a scenario where the PIA can be clearly attributed to one specific contributor (such as a distinct liquid cloud layer). Such a situation seems to be present during 245 the morning hours (4-5 UTC), when some low level, non-precipitating clouds are visible in the radar reflectivity time-height plot (indicated by the first square in Fig. 4a).
When zooming into this period (Fig. 5a), two main layers of cumuliform clouds clearly appear to be distinct from the upper ice cloud. The banded DFR signature visible in the upper ice cloud seems to be related to the presence of these shallow clouds. Excluding the parts of the cloud which clearly contain non-Rayleigh scattering particles, there isn't any perceptible 250 DFR increase in the ice cloud, which indicates that the majority of the attenuation signal is caused by the lower level clouds. The co-located ceilometer shows typical strong backscattering signals at cloud bottom combined with full extinction of the lidar signal above, which is a typical signature of liquid clouds (not shown). Even though the top of some of the cumuliform clouds reaches temperature slightly below 0°C, the clouds are assumed to be purely liquid. This is also confirmed by the slight increase of reflectivity with height, which is expected for an adiabatically increasing liquid water content. An opposite 255 reflectivity gradient would be expected for ice containing or mixed-phase clouds.
If these assumptions are correct and the attenuation signal is mainly caused by the lower level clouds, there should be a high correlation between the derived ∆PIA and the liquid water path (LWP) derived from a co-located microwave radiometer.
According to the processing steps described in Section 3, the filtered DFR depicted in Fig. 2 is obtained by screening out areas which indicate problematic radar volume matching (for example, close to cloud edges) or non-Rayleigh scattering particles 260 (highlighted by gray shading in Fig. 5b). As indicated by the black contour line in Fig. 5b, the Rayleigh plateau method is able to make use of a much thicker part of the ice cloud (up to 5 km) for estimating ∆PIA as compared to the Z e threshold method (Z Ka < −10 dBZ). The threshold method would only use the uppermost 1 km of the ice cloud, which includes cloud areas that are prone to volume matching issues and increasingly affected by the different sensitivity limits of the radars.
The derived ∆PIA is shown in Fig. 5c together with the LWP obtained from the co-located Humidity and Temperature  band using a statistical retrieval similar to Löhnert and Crewell (2003). Data gaps in LWP retrievals are due to regular MWR calibration procedures and intermediate azimuth/elevation scans. Strong and sharp LWP variations are found at the scale of less than a minute. They are clearly correlated with the presence of high reflectivity low-level cumulus clouds (auxilliary measurements from a ceilometer before and after the low cumulus clouds confirm that the upper cloud is composed of ice 270 only) and with the ∆PIA variations. In order to avoid discrepancies due to too long time averaging, a relatively high time resolution of 20 s is chosen for the retrieval of ∆PIA (Section 3) in order to account for the fast LWP variations in the observed cumuliform clouds.
The agreement between the time series of MWR derived LWP and ∆PIA (Fig. 5c) is remarkable. In the scatter plot of LWP and ∆PIA (Fig. 6), the points from cumuliform clouds during the morning hours follow closely the relation predicted by liquid 275 water refractive index models for a temperature of 0°C, i.e. about 7 dB two-way K a -W differential attenuation per kg m −2 of LWP. These results suggest that LWP larger than roughly 100 g m −2 can be retrieved with ∆PIA method provided that the PIA is due to liquid cloud water only. As seen in Fig. 6, no LWP is derived during rainy periods (before 01:00, between 07:00 and 08:00, 9:00 and 13:00 and after 16:00 UTC) because the presence of rain drops violates the MWR retrieval assumptions (scattering effects are assumed to be negligible). Nevertheless, the retrieved ∆PIA during these periods might be useful because 280 it can provide information on the integrated amount of rain and help to constrain radar retrievals.

Observed temperature dependence of differential attenuation due to liquid water
The close relation between ∆PIA and LWP shown in Fig. 6 seems to significantly change for the later period between 15:30 and 16:00 UTC (red points). The case overview (Fig. 4) shows that the low-level liquid clouds have mostly disappeared but the thick ice cloud has separated into two distinct layers with the upper one between 7 and 10 km and the lower one between 4 and 285 6 km. Interestingly, while the DFR structure of the upper layer appears to be similar to the earlier periods, the DFR in the lower ice cloud is extremely noisy. By experience with other instances, it is suspected that the radar beam mismatch is amplified in this cloud layer due to high spatio-temporal variability inside this cloud (for example caused by wind shear, convection, or turbulence).
14 https://doi.org/10.5194/amt-2020-159 Preprint. Discussion started: 5 May 2020 c Author(s) 2020. CC BY 4.0 License. A zoomed view of this time period (Fig. 7) reveals indeed a very high variability of reflectivity and DFR in time and space in 290 the ice cloud between 4 and 6 km. The signatures appear to be similar to generating cells (Kumjian et al., 2014) which are often observed as a result of instabilities at the top of ice clouds. When looking more carefully, a high correlation is revealed between the columns of enhanced reflectivity and both ∆PIA and LWP. It appears to be quite likely that supercooled liquid is generated by updraughts in this columns even though the temperature level is ≈-25°C. This lower temperature also explains well the different curve of the ∆PIA and LWP for this period in Fig. 6. The colder temperature causes the K a -W differential attenuation 295 produced by liquid droplets to be much smaller for the same LWP (see Section 2). When plotting the model prediction for ∆PIA and LWP for the lower temperature range (Fig. 8), the retrieved LWP and ∆PIA are fairly consistent with the liquid water refractive index models. In fact, such co-located MWR and K a -W radar observations would allow under ideal conditions a determination of the mean liquid water temperature. A similar rationale has been used by Matrosov and Turner (2018) with only a MWR with channels in the K a and W-bands. The radar approach provides of course more profile information which a 300 pure MWR method is unable to capture.
With this case study example, the objective was mainly to test how reliably and consistently the ∆PIA can be retrieved by the Rayleigh plateau method. Although situations with ice cloud above shallow liquid layers might not be uncommon at many sites, the origin of the ∆PIA signal is in general more complex and due to various sources, which will be investigated in the following section. Regarding a potential profiling of liquid water inside for example mixed-phase clouds, note that the 305 15 https://doi.org/10.5194/amt-2020-159 Preprint. Discussion started: 5 May 2020 c Author(s) 2020. CC BY 4.0 License. temperature sensitivity of ∆PIA is expected to be much stronger when using radar channels in the G-band (note the steep decrease of the purple lines at sub-freezing temperatures in Fig. 1). The recent realisation of G-Band radars  bears therefore a high potential for profiling liquid water when combined with low-frequency MWR.

Mixed-phase clouds and intense snowfall
The Rayleigh plateau method is now tested on a more complex case which comprises mixed ice and supercooled liquid as 310 well as snowfall on the ground. This case is characterized by a frontal passage that occurred on 21st February 2014 at the Hyytiälä field site, Finland (Fig. 9). The multi-frequency radar and auxiliary dataset have been recorded in the framework of the Biogenic Aerosols Effects on Clouds and Climate (BAECC) field experiment (Petäjä et al., 2016), during which the U.S. In this work, reflectivity data from the K a -band ARM zenith radar (KAZR; Isom et al. (2014b), Fig. 9a) and the marine W-band ARM cloud radar (MWACR; Isom et al. (2014a)) have been corrected for gas attenuation and re-gridded on a common time-height grid after compensation of mismatches in time and range. Indeed, using the cross-correlation of the reflectivity fields, it was found that best matching was obtained using offsets in range (30 m, i.e., one range gate) and time (0 to 4 s 320 between 0 and 5 km and 4 s above). The KAZR highest sensitivity (KAZRMD) and the general (KAZRGE) modes are properly inter-calibrated and then merged in order to maximise the signal to noise ratio (SNR) and avoid receiver saturation close to the ground (below approximately 1 km). In Fig. 9b, the DFR with SNR larger than -16 and -17.5 dB is depicted for KAZR and MWACR, respectively.

Department of Energy
This case represents a typical mixed-phase cloud, with persistent supercooled liquid layers (between the surface and 5 km) 325 leading to significantly rimed snow and graupel at the ground, as confirmed by the large bulk particle density (comprised between 200 and 500 kg m −3 ) retrieved from in-situ (Moisseev et al., 2017) and multi-frequency radar retrieval (Mason et al., 2018). Conversely, the period with stark reflectivities of more than 10 dBZ up to 6 km altitude just before midnight (Fig. 10a) was found to be dominated by large aggregates and snow rate up to 4 mm h −1 . A detailed analysis of this time period in Kalesse et al. (2016) shows that the liquid topped mixed-phase cloud starts being seeded around 23:00 UTC by ice falling from 330 the upper cirrus cloud. During this time period, the lower lidar backscatter (see Fig. 4c in Moisseev et al. (2017)) reveals that intense seeder-feeder process depletes supercooled water, which leads to a reduction of riming.
Similarly to the TRIPEX case, differential scattering due to large snowflakes is obvious in the spatially inhomogeneous DFR structures in the lower cloud layer (below 4 km in Fig. 9b). Furthermore, periods of significant differential attenuation can be identified as vertical lines of enhanced DFR in the cirrus cloud (e.g. around 22:00, 23:30 and 2:30 UTC), which is expected 335 to be composed of small ice crystals only. Again, the ∆PIA has been derived from the DFR Rayleigh plateau method and the following analysis will focus on this specific time period (dashed-line square in Fig. 9). Data after 3:00 UTC show high DFR (larger than 12 dB) due to the considerable non-Rayleigh scattering and attenuation produced by rain and melting ice (and, possibly, by radome attenuation). Though this information could be very useful in a full-column precipitation retrieval, profiles after 3:00 UTC are not further considered in this analysis because of the complexity of attenuation sources.

Path-integrated attenuation due to liquid cloud and snow
In the previous case, the lower level liquid cloud could be identified to be the major contributor to the attenuation signal. In this case, however, the possibility of attenuation due to ice and snow must also be considered as discussed in Section 2. A potential snow and ice attenuation signal should be detectable by comparing ∆PIA to the LWP measured by collocated MWR, which is insensitive to snow scattering at the low frequency bands used.

345
Similarly to the TRIPEX case, areas with high inhomogeneity and non-Rayleigh scattering are successfully filtered out by the DFR Rayleigh plateau method (gray shading in Fig. 10b). In this case, using the Z Ka < −10 dBZ threshold would retain approximately the same extent of data for deriving ∆PIA. However, Z Ka at the top of the lower cloud layer often satisfies the -10 dBZ threshold while the corresponding large DFR has clearly a non negligible contribution from non-Rayleigh scattering   In Fig. 10c, ∆PIA obtained from the filtered DFR is compared to the LWP measured by the collocated ARM microwave two-channel (MWR, 23.8 and 31.4 GHz, Cadeddu and Ghate (2014b)) and three-channel (MWR3C, 23.8 and 30 and 89 GHz, Cadeddu and Ghate (2014a)) radiometers. Except for the high reflectivity period between 23:00 and 0:00 UTC, a fairly good agreement is found between the timeseries of ∆PIA and LWP suggesting that most of the differential attenuation is produced 355 by cloud liquid water. Similarly to the TRIPEX case, strong variations on timescales of less than a minute can be found in the LWP and, consequently, on the DFR timeseries (e.g. around 1:00 UTC).
In comparison with the relations predicted by liquid water refractive index models between -15 and -5°C (i.e., for temperatures corresponding to the heights where the supercooled liquid water clouds are expected), the retrieved ∆PIA is slightly too large in reference to the two-channel LWP (Fig. 11). This small overestimation is particularly visible when reflectivity is high 360 (e.g. around 21:30, 0:20 and 2:30 UTC) and may be due to snow attenuation, as shown in the next section. Note that for this temperature range, the liquid water refractive index models slightly disagree: at -10°C, the Ellison (2007) model predicts 7 dB two-way K a -W differential attenuation per kg m −2 of liquid water while both Rosenkranz (2015) and Turner et al. (2016) models only predict 6 dB. Although the Turner et al. (2016) model was specifically developed for temperatures as low as -32°C, the radar data seem to agree better with the Ellison (2007) model, which predicts larger ∆PIAs for the same LWP. However, no definitive conclusions can be drawn about refractive index models at sub-freezing temperatures because of the presence of not well-quantified additional sources of attenuation (e.g. from snow).

Path-integrated differential attenuation due to snow
Attenuation due to snow and ice depends mainly on the total ice mass in the column but larger sizes and higher degree of riming are expected to further enhance the attenuation signal for a given snow mass (Battaglia et al., 2020a). This effect can be 370 seen during the high reflectivity period (between 23:00 and 0:00 UTC), where ∆PIA is found to reach up to 6 dB, while LWP remains low (Fig. 10c). ∆PIA variations during this time period seem correlated with the radar reflectivity field, suggesting that the differential attenuation is produced by the intense snow composed of large aggregates (snow rate of 4 mm h −1 and median mass diameters up to 5 mm, as retrieved from co-located ground-based in-situ instruments by Moisseev et al. (2017)).
In order to separate ∆PIA due to liquid and snow, the contribution of the MWR-retrieved LWP to ∆PIA is calculated using 375 the refractive index model from Turner et al. (2016). The difference to the total measured ∆PIA can then be assigned to snow attenuation. As a consistency check, the ice water content (IWC) is derived from Z Ka in dBZ using the relation proposed by Protat et al. (2007) for mid-latitudes (log 10 (IWC) = 0.000372Z Ka T + 0.0782Z Ka − 0.0153T − 1.54 with T the temperature in°C, and integrated over the altitude in order to obtain the ice water path (IWP). The resulting timeseries of snow attenuation and IWP are generally well correlated (Fig. 12). In particular, the peak of large snow attenuation deduced from the mismatch 380 between LWP and ∆PIA can be well explained by the corresponding large IWP. The presence of large snowflakes at 23:30 UTC is also supported by the disagreement between the MWR and MWR3C retrievals in Fig. 10c: while brightness temperatures at 30 GHz and below are relatively flat around this time, a slight enhancement of 6 K is observed at 89 GHz (not shown). Such a behaviour is consistent with an enhanced scattering produced by large snowflakes (Kneifel et al., 2010). The snow and ice 20 https://doi.org/10.5194/amt-2020-159 Preprint. Discussion started: 5 May 2020 c Author(s) 2020. CC BY 4.0 License. present in this case produced roughly 1.2 dB two-way attenuation per kg m −2 . These values are in agreement with self-similar Rayleigh-Gans attenuation computations for low-density aggregates of characteristic size equal to 5 mm (Fig. A6 in Battaglia et al. (2020a)). While this seems lower than the attenuation measured by Nemarich et al. (1988) using a horizontal link, this is larger than what is obtained when using the more recent relations found by Matrosov (2007)  Finally, for short time periods in Fig. 12, the disagreement between the total ∆PIA and liquid attenuation suggests significant 390 snow attenuation estimate while the IWP remains very low (e.g. around 22:00 and after 0:30 UTC). Interestingly, this corresponds to periods where very high LWPs were measured by MWRs (LWP > 0.6 kg m −2 in Fig. 10c). Such high LWPs are at the edge of the range of applicability of MWR retrievals. In fact, these conditions are likely favorable for the formation of drizzle drops which lead to larger differential attenuation per unit mass (section 2), thus violating the MWR retrieval assumptions.
High LWPs are also conducive for riming and heavily rimed snowflakes, which are very efficient in producing attenuation at 395 W-band (Battaglia et al., 2020a). Both circumstances can explain the disagreement observed between the retrieved ∆PIA and LWP.

Conclusions
Multi-frequency radar retrievals often require as important integral constraint a reliable estimate of the differential path integrated attenuation (∆PIA) caused by gases, rain, melting hydrometeors, cloud liquid water, and snow. While ∆PIA can be 400 relatively easy derived from nadir pointing radars, it is common practice for ground-based radars to derive ∆PIA by matching reflectivities at cloud top, where only Rayleigh targets with identical effective reflectivities (Z e ) are expected.
While this method works in many situations, it also has inherent problems: low concentrations of medium sized (non-Rayleigh) particles might produce Z e below the threshold, but they would be inappropriate for the Z e matching. Conversely, a high concentration of small particles could exceed the threshold, while still being valid Rayleigh-scattering particles. Finally, 405 the transition from Rayleigh to non-Rayleigh scatterers does not only depend on particle size but also on radar frequency which makes definition of Z e thresholds even more ambiguous. In the new approach presented in this work, the aim was instead to identify signatures of negligible particle growth in the vicinity of cloud top. For this, the key approach is searching for a Rayleigh plateau, i.e. a large enough region where the vertical gradient of DFR remains below a certain threshold. With this method, the region from which the ∆PIA can be derived is usually substantially enlarged compared to the Z e threshold method.

410
It also provides an indication of the quality of the DFR (e.g., appropriate beam alignment), which is usually not addressed by the Z e threshold method.
The new methodology is applied to two mid-latitude case studies representing different complexity of clouds. With a distinct low-level liquid cloud and a thick ice cloud aloft, the first case represents an ideal scenario to test the method, because the attenuation can be solely attributed to the lower liquid cloud. The comparison of the derived ∆PIA with time series of LWP 415 from co-located MWR shows a remarkable agreement, sometimes even for LWP values lower than 100 g m −2 . The second case represents a more commonly observed complex cloud occurrence including mixed-phase clouds and intense snowfall. Also in this case, from the comparison with the MWR, the ∆PIA is predominantly produced by cloud liquid. In the absence of a MWR, the ∆PIA method appears to be an alternative approach to retrieve LWP in situations where other sources of attenuation can be assumed to be negligible. If a co-located low frequency (f <90 GHz) MWR is available, the ∆PIA technique can also be used 420 to infer the approximate height of the liquid layer due to dependence of the liquid attenuation on temperature. During snowfall events or thick ice clouds without a melting layer or rain, the comparison of ∆PIA and LWP can be used to infer the attenuation signal caused by the frozen hydrometeors in the column. In the case study analyzed, the ∆PIA is generally very consistent with a radar derived ice water path. Noticeably, a deep snow system produces as much as 5 dB K a -W differential attenuation, which corresponds to an approximately snow attenuation coefficient of 1.2 dB attenuation per kg m −2 for this specific event.

425
Such values are within the range of the few available relations in the literature. However, the differences between the previously published relations might indicate a large dependence on the properties of snow particles (e.g. size, rimed mass fraction), which needs to be investigated on a larger data set.
In future work, this procedure will be systematically applied to a growing data set of K a -W band radar and MWR observations in order to thoroughly characterize snow attenuation at W-band, a key parameter for the retrieval of snow properties 430 from space-borne radars and MWRs. Quality-controlled smooth DFR profiles, a by-product of the technique, could also help to improve microphysical process studies.
In order to further disentangle the differential attenuation and scattering signal, the analysis of the multi-frequency Doppler spectra will be necessary. While several studies looking at rain and melting layer made significant progress in this direction, they also found that the quality and in particularly, the requirement on radar volume matching, are very high. The incorporation 435 of Doppler spectra in combination with newly developed G-Band radars is expected to bear great potential for profiling liquid water and snow even within thick mixed-phase clouds.